Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map

O. B. Isaeva ; M. A. Obychev ; D. V. Savin

Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 331-348.

Résumé

An abstract discrete time dynamical system, given by an implicit function of the values of a variable at successive moments of time, is presented. The dynamics of this system is defined ambiguously both in reverse and forward time. An example of a system of such type is described in the works of Bullett, Osbaldestin and Percival [Physica D, 1986, vol. 19, pp. 290–300; Nonlinearity, 1988, vol. 1, pp. 27–50]; it demonstrates some features of the behavior of Hamiltonian systems. The map under study allows a smooth transition from the case of the explicitly defined evolution operator to an implicit one and, further, to the “conservative” limit, corresponding to the symmetric evolution operator satisfying the unitarity condition. Being created on the basis of the complex Mandelbrot map, it demonstrates the transformation of the phenomena of complex analytical dynamics to “conservative” phenomena and allows us to identify the relationship between them.

Détail

Keywords: Mandelbrot set, Julia set, conservative and quasi-conservative dynamics, multistability
Mots-clés : implicit map.

Bibliographie

[1] Beck Ch., “Physical meaning for Mandelbrot and Julia sets”, Phys. D, 125:3–4 (1999), 171–182 | DOI | MR | Zbl

[2] Bohr T., Cvitanović P., Jensen M. H., “Fractal «aggregates» in the complex plane”, Europhys. Lett., 6:5 (1998), 445–450 | DOI

[3] Bullett Sh. R., Osbaldestin A. H., Percival I. C., “An iterated implicit complex map”, Phys. D, 19:2 (1986), 290–300 | DOI | MR | Zbl

[4] Bullett Sh., “Dynamics of quadratic correspondences”, Nonlinearity, 1:1 (1988), 27–50 | DOI | MR | Zbl

[5] Cayley A., “Desiderata and suggestions: No. 3. The Newton – Fourier imaginary problem”, Am. J. Math., 2:1 (1879), 97 | DOI | MR

[6] Chen C., Györgyi G., Schmidt G., “Universal transition between Hamiltonian and dissipative chaos”, Phys. Rev. A, 34:3 (1986), 2568–2570 | DOI | MR

[7] Cvitanović P., Myrheim, J., “Universality for period $n$-tuplings in complex mappings”, Phys. Lett. A, 94:8 (1983), 329–333 | DOI | MR

[8] DiFranco D. E., Cham T.-J., Regh J. M., “Reconstruction of $3$D figure motion from $2$D correspondences”, Proc. of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, v. 1, IEEE Computer Society Press, Los Alamitos, 2001, 307–314

[9] Éntin M. V., Éntin G. M., “Scale invariance in percolation theory and fractals”, JETP Lett., 64:6 (1996), 467–472 | DOI

[10] Fatou P., “Sur les équations fonctionnelles”, Bull. Soc. Math. France, 47 (1919), 161–271 ; Fatou P., “Sur les équations fonctionnelles”, Bull. Soc. Math. France, 48 (1920), 208–314 | DOI | MR | Zbl | DOI | MR

[11] Feigenbaum M. J., “Universal behavior in nonlinear systems”, Phys. D, 7:1–3 (1983), 16–39 | DOI | MR

[12] Feigenbaum M. J., Kadanoff L. P., Shenker S. J., “Quasiperiodicity in dissipative systems: A renormalization group analysis”, Phys. D, 5:2–3 (1982), 370–386 | DOI | MR

[13] Feudel U., Kuznetsov S., Pikovsky A., Strange nonchaotic attractors: Dynamics between order and chaos in quasiperiodically forced systems, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 56, World Sci., Hackensack, N.J., 2006, 228 pp. | MR

[14] Feudel U., “Complex dynamics in multistable systems”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18:6 (2008), 1607–1626 | DOI | MR

[15] Gardini L., Hommes C., Tramontana F., de Vilder R., “Forward and backward dynamics in implicitly defined overlapping generations models”, J. Econ. Behav. Organ., 71:2 (2009), 110–129 | DOI

[16] Govin M., Jauslin H. R., Cibils M., “Julia sets in iterative KAM methods for eigenvalue problems”, Chaos Solitons Fractals, 9:11 (1998), 1835–1846 | DOI | MR | Zbl

[17] Gol'berg A. I., Sinai Ya. G., Khanin K. M., “Universal properties for sequences of bifurcations of period three”, Russian Math. Surveys, 38:1 (1983), 187–188 | DOI | MR | Zbl

[18] Hill D. L., “Control of implicit chaotic maps using nonlinear approximations”, Chaos, 10:3 (2000), 676–681 | DOI | MR | Zbl

[19] Isaeva O. B., Eliseev M. V., Rozhnev A. G., Ryskin N. M., “Simulation of field emission from fractal surface”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 7:5 (1999), 33–43 (Russian)

[20] Isaeva O. B., Kuznetsov S. P., “On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade”, Regul. Chaotic Dyn., 5:4 (2000), 459–476 | DOI | MR | Zbl

[21] Isaeva O. B., Kuznetsov S. P., Ponomarenko V. I., “Mandelbrot set in coupled logistic maps and in an electronic experiment”, Phys. Rev. E., 64:5 (2001), 055201(R), 4 pp. | DOI

[22] Isaeva O. B., “On possibility of realization of the phenomena of complex analytical dynamics for the physical systems, built up of coupled elements, which demonstrate period-doublings”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 9:6 (2001), 129–146 (Russian) | Zbl

[23] Isaeva O. B., Kuznetsov S. P., Osbaldestin A. H., “A system of alternately excited coupled non-autonomous oscillators manifesting phenomena intrinsic to complex analytical maps”, Phys. D, 237:7 (2008), 873–884 | DOI | MR | Zbl

[24] Julia G., “Mémoire sur l'itération des fonctions rationnelles”, J. Math. Pure Appl. (8), 1 (1918), 47–246

[25] Kaneko K., Collapse of tori and genesis of chaos in dissipative systems, World Sci., Singapore, 1986, 264 pp. | MR | Zbl

[26] Kaneko K., Collapse of tori and genesis of chaos in dissipative systems, World Sci., Singapore, 1986, xii+264 pp. | MR | Zbl

[27] Kuznetsov S. P., Dynamical chaos and uniformly hyperbolic attractors: From mathematics to physics, R Dynamics, Institute of Computer Science, Izhevsk, 2013 (Russian)

[28] Lee T. D., Yang C. N., “Statistical theory of equations of state and phase transitions: 2. Lattice gas and Ising model”, Phys. Rev. (2), 87:3 (1952), 410–419 | DOI | MR | Zbl

[29] Mandelbrot B. B., The fractal geometry of nature, Freeman, San-Francisco, Calif., 1982, 460 pp. | MR | Zbl

[30] Mandelbrot B. B., Fractals and chaos: The Mandelbrot set and beyond, Springer, New York, 2004, XII+308 pp. | MR | MR | Zbl

[31] MacKay R. S., van Zeijts J. B. J., “Period doubling for bimodal maps: A horse-shoe for a renormalization operator”, Nonlinearity, 1:1 (1988), 253–277 | DOI | MR | Zbl

[32] Encyclopaedia of mathematics, v. 1 (A–B), ed. M. Hazewinkel, Kluwer, Dordrecht, 1987, xi + 488 pp. | MR

[33] Mestel B. D., Osbaldestin A. H., “Renormalisation in implicit complex maps”, Phys. D, 39:2–3 (1989), 149–162 | DOI | MR | Zbl

[34] Milnor J. W., Dynamics in one complex variable: Introductory lectures, Vieweg, Teubner, Wiesbaden, 2000, 251 pp. | Zbl

[35] Peitgen H.-O., Richter P. H., The beauty of fractals: Images of complex dynamical systems, Springer, Berlin, 1986, 202 pp. | MR | Zbl

[36] Peckham B. B., Montaldi J., “Real continuation from the complex quadratic family: Fixed-point bifurcation sets”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10:2 (2000), 391–414 | DOI | MR | Zbl

[37] Pecora N., Tramontana F., “Maps with vanishing denominator and their applications”, Front. Appl. Math. Stat., 2 (2016), 11 | DOI

[38] Pikovsky A., Rosenblum M., Kurths J., Synchronization: A universal concept in nonlinear sciences, Cambridge Nonlinear Sci. Ser., 12, Cambridge Univ. Press, Cambridge, 2001, 432 pp. | MR | Zbl

[39] Pomeau Y., Manneville P., “Intermittent transition to turbulence in dissipative dynamical systems”, Comm. Math. Phys., 74:2 (1980), 189–197 | DOI | MR

[40] Prigogine I., Stengers I., The end of certainty, Free Press, New York, 1997, 240 pp.

[41] Reichl L. E., The transition to chaos in conservative classical systems: Quantum manifestations, Springer, New York, 1992, 551 pp. | MR | Zbl

[42] Roberts J. A. G., Quispel G. R. W., “Chaos and time-reversal symmetry: Order and chaos in reversible dynamical systems”, Phys. Rep., 216:2–3 (1992), 63–177 | DOI | MR

[43] Ruelle D., Takens F., “On the nature of turbulence”, Comm. Math. Phys., 20:3 (1971), 167–192 | DOI | MR | Zbl

[44] Sclaroff S., Pentland A., “Generalized implicit functions for computer graphics”, Comput. Graph., 25:4 (1991), 247–250 | DOI

[45] Schuster H. G., Just W., Deterministic chaos: An introduction, Wiley-VCH, Weinheim, 2005, 312 pp. | MR | Zbl

[46] Shudo A., Ishii Y., Ikeda K. S., “Julia set describes quantum tunnelling in the presence of chaos”, J. Phys. A, 35:17 (2002), L225–L231 | DOI | MR | Zbl

[47] Widom M., “Renormalization group analysis of quasiperiodicity in analytic maps”, Comm. Math. Phys., 92:1 (1983), 121–136 | DOI | MR | Zbl

[48] Vlasenko I. Yu., Internal maps: Topological invariants and their applications, Ukrainian Acad. Sci., Kiev, 2014, 214 pp.

[49] Yang C. N., Lee T. D., “Statistical theory of equations of state and phase transitions: 1. Theory of condensation”, Phys. Rev. (2), 87:3 (1952), 404–409 | DOI | MR | Zbl

[50] Zisook A. B., “Universal effects of dissipation in two-dimensional mappings”, Phys. Rev. A, 24:3 (1981), 1640–1642 | DOI | MR